Research Article Existence of Positive Solution to Second-Order Three-Point BVPs on Time Scales

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1 Hindwi Publishing Corportion Boundry Vlue Problems Volume 2009, Article ID , 6 pges doi: /2009/ Reserch Article Existence of Positive Solution to Second-Order hree-point BVPs on ime Scles Jin-Ping Sun Deprtment of Applied Mthemtics, Lnzhou University of echnology, Lnzhou, Gnsu , Chin Correspondence should be ddressed to Jin-Ping Sun, jpsun@lut.cn Received 19 April 2009; Accepted 14 September 2009 Recommended by Knishk Perer We re concerned with the following nonliner second-order three-point boundry vlue problem on time scles x ΔΔ t f t, x t,, b, x 0, x σ 2 b δx η, where, b with <b, η, b nd 0 <δ< σ 2 b / η. A new representtion of Green s function for the corresponding liner boundry vlue problem is obtined nd some existence criteri of t lest one positive solution for the bove nonliner boundry vlue problem re estblished by using the itertive method. Copyright q 2009 Jin-Ping Sun. his is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. 1. Introduction Let be time scle, tht is, is n rbitrry nonempty closed subset of R. For ech intervl I of R, we define I I. For more detils on time scles, one cn refer to 1 5. Recently, three-point boundry vlue problems BVPs for short for second-order dynmic equtions on time scles hve received much ttention. For exmple, in 2002, Anderson 6 studied the following second-order three-point BVP on time scles: u Δ t t f u t 0, 0,, u 0 0, u αu ( η ), 1.1 where 0,, η 0,ρ nd 0 <α</η. Some existence results of t lest one positive solution nd of t lest three positive solutions were estblished by using the well-known Krsnoselskii nd Leggett-Willims fixed point theorems. In 2003, Kufmnn 7 pplied the Krsnoselskii fixed point theorem to obtin the existence of multiple positive solutions to the BVP 1.1. For some other relted results, one cn refer to 8 10 nd references therein.

2 2 Boundry Vlue Problems In this pper, we re concerned with the existence of t lest one positive solution for the following second-order three-point BVP on time scles: x ΔΔ t f t, x t,, b, ( ) x 0, x σ 2 b δx ( η ). 1.2 hroughout this pper, we lwys ssume tht, b with <b, η, b,nd0<δ< σ 2 b / η. It is interesting tht the method used in this pper is completely different from tht in 6, 7, 9, 10, tht is, new representtion of Green s function for the corresponding liner BVP is obtined nd some existence criteri of t lest one positive solution to the BVP 1.2 re estblished by using the itertive method. For the function f, we impose the following hypotheses: H1 f :, b R R is continuous; H2 for fixed, b, f t, u is monotone incresing on u; H3 there exists q 0, 1 such tht f t, ru r q f t, u for r 0, 1, t, u, b R. 1.3 Remrk 1.1. If H3 is stisfied, then f t, λu λ q f t, u for λ 1,, t, u, b R Min Results Lemm 2.1. he BVP 1.2 is equivlent to the integrl eqution x t K t, s f s, x s Δs,, σ 2 b, 2.1 where δg ( η, s ) K t, s G t, s σ 2 b δ ( ) t η 2.2 is clled the Green s function for the corresponding liner BVP, here 1 t ( σ 2 b σ s ), t s, G t, s σ 2 b σ s ( σ 2 b t ), t σ s 2.3

3 Boundry Vlue Problems 3 is the Green s function for the BVP: x ΔΔ t 0,, b, ( ) x x σ 2 b Proof. Let x be solution of the BVP: x ΔΔ t f t, x t,, b, ( ) x x σ 2 b hen, it is esy to know tht x t G t, s f s, x s Δs, x x ( ) σ 2 b 0., σ 2 b, 2.6 Now, if x is solution of the BVP 1.2, then it cn be expressed by x t C 1 C 2 t x t, 2.7 which together with the boundry conditions in 1.2 nd 2.6 implies tht x t δx ( η ) σ 2 b δ ( η ) t x t K t, s f s, x s Δs,, σ 2 b. 2.8 On the other hnd, if x stisfies 2.1, then it is esy to verify tht x is solution of the BVP 1.2. Lemm 2.2. For ny t, s, σ 2 b, σ b, one hs δg ( η, s ) σ 2 b δ ( δg ( η, s ) ] ) t K t, s 1 η σ 2 b δ ( η ) t. 2.9 Proof. Since it is obvious from the expression of G t, s tht 0 G t, s 1 ( ) σ 2 b t σ 2 b t, t, s, σ 2 b, σ b, 2.10 we know tht 2.9 is fulfilled.

4 4 Boundry Vlue Problems Our min result is the following theorem. heorem 2.3. Assume tht (H1) (H3) re stisfied. hen, the BVP 1.2 hs t lest one positive solution w. Furthermore, there exist M m>0 such tht m t w t M t,, σ 2 b Proof. Let } E x x :, σ 2 b ] R is continuous, } D x E there exist M x m x >0 such tht m x t x t M x t for t, σ 2 b, } P x E x t 0fort, σ 2 b Define n opertor F : D P: Fx t K t, s f s, x s Δs,, σ 2 b hen it is obvious tht fixed points of the opertor F in D re positive solutions of the BVP 1.2. First, in view of H2, it is esy to know tht F : D P is incresing. Next, we my ssert tht F : D D, which implies tht for ny x D, there exist positive constnts l nd L such tht Fx t Lx t, Fx t lx t for x, σ 2 b In fct, for ny x D, there exist 0 <m x < 1 <M x such tht m x t x t M x t for, σ 2 b, 2.15 which together with H2, H3,ndRemrk 1.1 implies tht m x q f t, t f t, x t M x q f t, t for, b By Lemm 2.2 nd 2.16, for ny, σ 2 b, we hve Fx t M x q δg ( η, s ) ] 1 σ 2 b δ ( η ) f s, s Δs t, Fx t m x q δg ( η, s ) σ 2 b δ ( )f s, s Δs t. η 2.17

5 Boundry Vlue Problems 5 If we let M 0 M x q δg ( η, s ) ] 1 σ 2 b δ ( η ) f s, s Δs, m 0 m x q δg ( η, s ) σ 2 b δ ( )f s, s Δs, η 2.18 then it follows from 2.17 nd 2.18 tht m 0 t Fx t M 0 t for, σ 2 b, 2.19 which shows tht Fx D. Now, for ny fixed h 0 D, we denote } l h0 sup l>0 Fh 0 t lh 0 t, t, σ 2 b 2.20 } L h0 inf L>0 Fh 0 t Lh 0 t, t, σ 2 b m min 2, l h 0 1/ 1 q }, M mx 2, L h0 1/ 1 q } 2.22 nd let u n t Fu n 1 t, v n t Fv n 1 t,, σ 2 b,n 1, 2,..., 2.23 where u 0 t mh 0 t, v 0 t Mh 0 t,, σ 2 b hen, it is esy to know from 2.20, 2.21, 2.22, 2.23, 2.24, H3, ndremrk 1.1 tht u 0 t u 1 t u n t v n t v 1 t v 0 t,, σ 2 b Moreover, if we let t 0 m/m, then it follows from 2.22, 2.23, 2.24,nd H3 by induction tht u n t t 0 qn v n t,, σ 2 b, n 0, 1, 2,..., 2.26 which together with 2.25 implies tht for ny positive integers n nd p, 0 u n p t u n t 1 t 0 qn] Mh 0 t,, σ 2 b. 2.27

6 6 Boundry Vlue Problems herefore, there exists w D such tht u n t } n 0 nd v n t } n 0 converge uniformly to w on, σ 2 b nd u n t w t v n t, Since F is incresing, in view of 2.28, we hve, σ 2 b,n 0, 1, 2, u n 1 t Fu n t Fw t Fv n t v n 1 t,, σ 2 b,n 0, 1, 2, So, Fw t w t,, σ 2 b, 2.30 which shows tht w is positive solution of the BVP 1.2. Furthermore, since w D, there exist M m>0 such tht m t w t M t,, σ 2 b Acknowledgment his work is supported by the Ntionl Nturl Science Foundtion of Chin References 1 R. P. Agrwl nd M. Bohner, Bsic clculus on time scles nd some of its pplictions, Results in Mthemtics, vol. 35, no. 1-2, pp. 3 22, M. Bohner nd A. Peterson, Dynmic Equtions on ime Scles: An Introduction with Appliction, Birkhäuser, Boston, Mss, USA, M. Bohner nd A. Peterson, Advnces in Dynmic Equtions on ime Scles, Birkhäuser, Boston, Mss, USA, S. Hilger, Anlysis on mesure chins unified pproch to continuous nd discrete clculus, Results in Mthemtics, vol. 18, no. 1-2, pp , V. Lkshmiknthm, S. Sivsundrm, nd B. Kymkcln, Dynmic Systems on Mesure Chins, vol. 370 of Mthemtics nd Its Applictions, Kluwer Acdemic Publishers, Dordrecht, he Netherlnds, D. R. Anderson, Solutions to second-order three-point problems on time scles, Journl of Difference Equtions nd Applictions, vol. 8, no. 8, pp , E. R. Kufmnn, Positive solutions of three-point boundry-vlue problem on time scle, Electronic Journl of Differentil Equtions, vol. 2003, no. 82, 11 pges, R. A. Khn, J. J. Nieto, nd V. Otero-Espinr, Existence nd pproximtion of solution of three-point boundry vlue problems on time scles, Journl of Difference Equtions nd Applictions, vol. 14, no. 7, pp , H. Luo nd Q. M, Positive solutions to generlized second-order three-point boundry-vlue problem on time scles, Electronic Journl of Differentil Equtions, vol. 2005, no. 17, 14 pges, H.-R. Sun nd W.-. Li, Positive solutions for nonliner three-point boundry vlue problems on time scles, Journl of Mthemticl Anlysis nd Applictions, vol. 299, no. 2, pp , 2004.